On Next-Nearest-Neighbor Interaction in Linear Chain. I

Ground‐state properties of the Hamiltonian $$\mathrm{H=}\frac{1}{2}J{\displaystyle \sum _{\text{i=1}}^N}{\sigma }_i{\mathrm{·\sigma }}_{\text{i+1}}+\frac{1}{2}\mathrm{J\alpha }{\displaystyle \sum _{\text{i=1}}^N}{\sigma }_i{\mathrm{·\sigma }}_{\text{i+2}}$$ (σ_N+1 ≡ σ₁, σ_N+2 ≡ σ₂) are studied for both signs of J and −1 ≤ α ≤ 1 to gain insight into the stability of the ground state with nearest‐neighbor interactions only (α = 0) in the presence of the next‐nearest‐neighbor interaction. Short chains of up to 8 particles have been exactly studied. For J > 0, the ground state for even N belongs always to spin zero, but its symmetry changes for certain values of α. For J < 0, the ground state belongs either to the highest spin (ferromagnetic state) or to the lowest spin and so to zero for even N. The trend of the results suggests that these facts are true for arbitrary N and that the critical value of α is probably zero. Upper and lower bounds to the ground‐state energy per spin of the above Hamiltonian are obtained. Such bounds can also be obtained for the square lattice with the nearest‐ as well as the next‐nearest‐neighbor interaction.